Robust Optimization-Based Generation Self-Scheduling under Uncertain Price

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Volume 2011, Article ID 497014,17pages doi:10.1155/2011/497014

Research Article

Robust Optimization-Based Generation Self-Scheduling under Uncertain Price

Xiao Luo,

¹

Chi-yung Chung,

¹

Hongming Yang,

²

and Xiaojiao Tong

³

1Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

2College of Electrical and Information Engineering, Changsha University of Science and Technology, Changsha 410004, China

3Department of Mathematics, Hengyang Normal University, Hengyang 421002, China

Correspondence should be addressed to Xiaojiao Tong,[email protected] Received 19 September 2010; Revised 20 November 2010; Accepted 24 January 2011 Academic Editor: J. Rodellar

Copyrightq2011 Xiao Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper considers generation self-scheduling in electricity markets under uncertain price. Based on the robust optimizationdenoted as ROmethodology, a new self-scheduling model, which has a complicated max-min optimization structure, is set up. By using optimal dual theory, the proposed model is reformulated to an ordinary quadratic and quadratic cone programming problems in the cases of box and ellipsoidal uncertainty, respectively. IEEE 30-bus system is used to test the new model. Some comparisons with other methods are done, and the sensitivity with respect to the uncertain set is analyzed. Comparing with the existed uncertain self-scheduling approaches, the new method has twofold characteristics. First, it does not need a prediction of distribution of random variables and just requires an estimated value and the uncertain set of power price. Second, the counterpart of RO corresponding to the self-scheduling is a simple quadratic or quadratic cone programming. This indicates that the reformulated problem can be solved by many ordinary optimization algorithms.

1. Introduction

Electricity market is a system which organizes, manages, and coordinates the power system by means of laws and economic tools under the principle of openness, competition, and fairness. The aim of electricity market is to improve the eﬃciency of power industry, to lower electric price, and to ensure the security of power system at the same time. The operation process of electricity market is as follows: firstly, the market participants submit their bids to the independent system operatorISO, considering their own profit maximization; secondly, under the power system security limits, the ISO decides the dispatch schedule such as the

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generated energy, load power, and spot price; thirdly, the market participants submit their bids of next period bids to ISO again. This process shows that for the generation company, it is important to make an available self-scheduling in the competitive environment so that the ISO can accept their bids. Finally, the aim of our research is to set up a suitable self-schedular model and to find an eﬀective solution approach.

In power system analysis, the generation self-scheduling is based on optimization approach. Two types of models are used for such problem. One is the determinate optimization approach, and the other is the uncertainty optimization approach with uncertain parameters. Since the later can describe the market operation action better, we will pay our attention to it in this paper. In the literature, the uncertain self-scheduling approach has been studied extensively from the model and algorithm aspects see 1–9. There are two available approaches to handle the uncertainty in competitive electricity markets:

probabilistic approach and fuzzy approach. By using the probabilistic method, Conejo et al.

studied the self-scheduling models of multiperiod schedule where the generation company is as price-takers2and a price-makers6in a pool-based electricity market, respectively.

Then a correspondent mixed-integer quadratic programming and a mixed-integer linear programming are set up. In their research, the profit and risk are considered simultaneously.

Furthermore, as an example, the hydro producer’s self-scheduling is considered with start-up costs1, and an 0/1 mixed-integer linear programming model is established. Yamin studied the self-scheduling model by using the fuzzy method8,9. A comprehensive fuzzy approach for self-scheduling problem is set up9, with the uncertainty of the demand, spinning and nonspinning reserves, price, and so forth. Recently, due to the nice mathematical property of conditional Value-at-RiskCVaR see10,11, Jabr3combined the CVaR method into generation self-scheduling, under the known random distribution of power price. Then a second-order cone program is established and the polynomial interior-point approach is adopted. Furthermore, considering the case that the mean vector and covariance matrix of probability distribution may be known partially, Jabr4,5developed the methodology3 and presented a worst-case robust profit model for the generation self-scheduling. The new model is reformulated as a symmetric cone optimization under special box uncertainty. For the multiperiod consideration, Tseng and zhu 7 studied the self-scheduling and bidding strategy of a thermal generation with the ramp constraints. All of these make a meaningful contribution for the ordinal market operation and provide a guideline of bidding strategy of generation companies.

We also note that the probabilistic approach is based on the known or partial known distribution of random variable, and a fuzzy approach is depend on the so- called membership function. However, in the complicated electricity market, the distribution prediction of random variable is diﬃcult, which may result an inaccurate forecast of random variables and yields a bad generation self-scheduling. This motivates us to find other method for the self-scheduling approach.

Mathematically, there is another typical approach for the uncertainty optimization problems, called robust optimizationRO. The RO method has been studied and applied various aspects recently see 11–15 and references therein. The main characteristic of RO approach solves the optimal problems based on a uncertain set of parameters, not the distribution of them. Such method can avoid the prediction of uncertain parameters in power markets and provide a motivation for the study of the generation self-scheduling.

Under the uncertain price, this paper addresses the self-scheduling via the RO approach. A min-max self-scheduling model is set up where the security constraints of the system are considered. In order to facilitate the model, we use the dual theory of

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optimization and obtain the correspondent counterpart. The reformulation is a typical quadratic optimization programming and can be solved easily. The characteristics of such research are twofold. First, the self-scheduling model does not need a distribution forecast of electricity price, but just need the possible set of the price, for example, a box region set.

This is the main characteristics of RO approach and the diﬀerence with3–5. Second, the counterpart of generation self-scheduling is ordinary convex quadratic and quadratic cone programming. This is very helpful from the viewpoint of computation.

The paper is organized as follows. Section 2 introduces the RO problem and its counterpart under a linear uncertain set. Section 3 sets up the RO-based self-scheduling model and facilitates the max-min optimization model with the cases of box and ellipsoidal uncertain sets. In Section 4, a numerical example of IEEE-30 system is tested, and some comparison with CVaR approach is also done. Last section addresses some conclusions.

2. Robust Optimization and Its Counterpart

This section presents the mathematical analysis for RO problem. The main objective is to transfer the uncertain optimization problem into a determinate optimization.

2.1. Robust Optimization

A general mathematical programming is of the form minx∈Rⁿ f₀x, ξ

s.t. f_ix, ξ≤0, ∀ξ∈ Ui1, . . . , m, 2.1 wherexis the designdecidedvector, the functionsf₀ :Rⁿ → Rthe objective function and fi : Rⁿ → R i 1, . . . , m are structural elements of the problemthe constrained functions, andξ∈ U ⊂R^lstands for the dadaor called parameters.

We have the following observation for problem2.1:

iif there is no parameter vectorξor the vectorξis fixedi.e.,Uhas finite points, the problem reduces to ordinary nonlinear programming problems;

iiifξ ∈ Uwith infinite elements, that is, the parameter vectorξbelongs to some set, then2.1is a uncertainty optimization.

The major challenges associated with above uncertain optimization are

iwhen and how can we reformulate or approximate 2.1 as a “computationally tractable” optimization problem?

iiHow to specify reasonable uncertainty setUin practical applications?

Typically, a min-max model is used to handle the model2.1, called robustor worst- caseversion, as follows:

minx∈Rⁿ sup

ξ∈U

f₀x, ξ

s.t. fix, ξ≤0, ∀ξ∈ Ui1, . . . , m

2.2

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or equivalently

minx∈Rⁿ sup

ξ∈U f₀x, ξ s.t. sup

ξ∈U

f_ix, ξ≤0, i1, . . . , m. 2.3

We call2.2and2.3robust optimizationROmodels. In this paper, we use2.3as a studied model. In what follows, we analyze its counterpart under some special uncertain setU.

2.2. Counterpart of RO under Linear Uncertain Set

We assumeUto be a linear version

UUτ ≡

ξ τDδ:δ_p≤1

, 2.4

whereξis the estimate value ofξ,τ ∈RandD∈R^l×l^δare the given constant and correlative matrix with respect to parameter vector ξ. The norm · _p is chosen as · ₁, · ₂, · _∞ norms.

Suppose that the functionsf₀, f_i i1, . . . , min2.3are continuously diﬀerentiable with respect toxandξ. Then we can facilitate the min-max model2.3.

(i) Objective functionf0

We make an approximation for the objective function atξ∈ Uτ as follows:

f₀

x,ξ τDδ

≈f₀ x,ξ

τ

∇ξf₀, Dδ

ξξ

f₀ x,ξ

τ

D^T∇ξf₀, δ

ξξ. 2.5

Herea, bmeans the internal product of vector. From the bounded property ofUτ and the formulaa, b≤ a_pb_q, we have the following derivation:

sup

ξ∈Uτ

f₀x, ξ max

ξ∈Uτ f₀x, ξ≈f x,ξ

τmax

δp≤1

D^T

∇ξf0, δ

ξξ

f₀ x,ξ

τ

D^T∇ξf₀

ξξ

_qf₀ x,ξ

τ

∂_ξf₀ D

ξξ_q,

2.6

whereqsatisfies 1/p 1/q1,D^Tis the adjoint ofD, and∇f ∂f^T.

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(ii) Constraint functionfi i1, . . . , m

Similarly, for the constraint: sup_u∈U_τf_i≤0, we have that

maxξ∈Uτ

f_ix, ξ≈fi

x,ξ τ

∂ξfi D

ξξ

q. 2.7

From2.6and2.7, we obtain the approximation problem of2.3underUU_τ: minu f0

x,ξ τ

∂ξf0

x,ξ D_q s.t. f_i

x,ξ τ

∂_ξf_i x,ξ

D_q≤0, i1, . . . , m. 2.8

We call2.8the counterpart of RO2.3.

Remark 2.1. iThe optimization problem of2.8is an ordinary optimization with the known estimate pointξ.

iiIffi i0,1, . . . , mis a linear function with respect to the parameter vectorξ, then the derivation is accurate, that is, the approximate equality becomes equality.

iiiThe RO approach can be extended for solving the following general optimization problem:

minx,u fx, u, ξ s.t. hx, u, ξ 0,

gx, u, ξ≤0,

2.9

where the variables x ∈ Rⁿ^x and u ∈ Rⁿ^u represent state variable and control variable, respectively.ξ∈R^lis the system parameterξ∈ Uτ defined in2.4.h:Rⁿ^xⁿ^uⁿ^l → Rⁿ^x.

Similar to Theorem 3.1 in16, we have the following feasibility with respect to the original uncertain optimization, which shows that the feasibility is controlled byτdefined in setU.

Theorem 2.2. Letxbe strictly feasible to2.8at pointξwithτ > 0. Assume that in the setUτ,

∇ξfix, ξis Lipschitz continuous with moduloL. Then it holds that

fix, ξ≤ L

2

D²τ², i1, . . . , m, ∀ξ∈ Uτ. 2.10

3. RO-Based Generation Self-Scheduling under Price Uncertainty

In this section, by using the RO method, we will set up the self-scheduling model under uncertain price. We call it RO-based generation self-scheduling throughout this paper.

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3.1. Self-Scheduling Problem in Power Systems

A generation self-scheduling can be defined by the following nonlinear programming:

maxPG∈ΠfPG, λ, 3.1

where the decision variable is the generation output PG, and λ is the power price. The objective and constraintsfeasible regionare defined as follows.

(i) Objective function

fPG, λrepresents the profitreturnof generating company

fPG, λ λ^TP_G−^N^G

i1

C_iPGi. 3.2

HereC_iPGiis the generator cost function, which is defined by a quadratic function

CiPGi ai biPGi ciP_Gi² fori1, . . . , NG. 3.3 (ii) Feasible regionΠ

The feasible regionΠofP_Gconsists of power generation limits, dc network model constraint, intact network line flow constraints, and security constraintssee3,17for the definition of Π:

1power generation limits

P_Gi^min ≤P_Gi≤P_Gi^max, 3.4

2DC network model

0≤P_i

j∈ki

a_ij δ_i−δ_j

≤P_Di, 3.5

3intact network line flow constraints

−T_ij^max≤T_ij−aij δ_i−δ_j

≤T_ij^max, 3.6

4security constraints following the outage of linesm₁k₁tom_rk_r in terms of flows in the intact network

−T_ij^max≤T_ij ^r

l1

ω_lT_m_l_k_l≤T_ij^max, 3.7

where the variables and parameters have the following meaning.

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kiis the set of nodes connected to nodei;Piis power injection at nodei0 orPGi; P_Gi^maxandP_Gi^minrepresent the maximum and minimum limit ofP_Gi;P_Diis the forecasted power demand at nodei;Tij expresses the intact power flow on lineij;Tij is the contingent power flow on lineij;T_ij^max: andT_ij^maxare the prefault and postfaultemergencyrating of lineij;Vi

denotes the voltage magnitude at nodei;a_ijis electrical susceptance;δ_iexpresses the voltage angle at nodei, δ₁0;ωlis thelth element of the row vector of load transfer coeﬃcients.

(iii) Power price

λ is the vector of locational marginal prices LMPs. Here we assume that the price is uncertain parameter with the following version:

λλ ζ 3.8

or

λλ Aζ, 3.9

where priceλis the given power price as an estimated value andζis a price fluctuation. The matrixAis an associated matrix of node price. The uncertain parameter of price is specialized byζ.

3.2. RO-Based Generation Self-Scheduling Model

According to the uncertain generation self-scheduling problem3.1, we will consider the robust or called worst-case version under some special set of ζ ∈ U. Note that in such model, the uncertain parameteri.e., priceis just in the objective with a linear version. We transfer the uncertain model3.1to a deterministic optimization by

maxPG

ζ∈UinffPG, λ s.t. PG∈Π.

3.10

Since the RO-based model3.10has a complicated min-max structure of optimization, we will facilitate the model and obtain the correspondent counterpart of RO problem. To this end, we consider two special cases of uncertain price setUin3.8and3.9as

U

ζ|e^Tζ0, ζ≤ζ≤ζ ,

U

ζ|e^TAζ0, ζ ≤1 ,

3.11

whereedenotes the vector of ones andζandζare given constant vectors. The above two sets are called box uncertainty and ellipsoidal uncertainty, respectively.

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Remark 3.1. Note that for solving the same generation self-scheduling, the main diﬀerence between our method and ones in 3–5 is twofold. First, our method is based on the RO approach and without the prediction of random variables. This is easily done in practical application, whereas the method in 3–5depends on the distribution of uncertain power price, which needs a forecast of uncertain price. Second, two methods have a diﬀerent focus on the problem. Our approach considers optimization under the worst-case, and3–5solved the problem under probability level of risk measure.

In the remainder of two subsections, our aim is to reformulate the optimization problem3.10to an ordinary optimization for cases of uncertain cases3.11.

3.3. Counterpart of RO-Based Self-Scheduling with Box Uncertainty

Since the setUis bounded, the objective function of3.10can be rewritten as

maxPG∈Πinf

ζ∈UfPG, λ max

PG∈Πmin

ζ∈U

λ ζT

PG−^N^G

i1

ai biPGi ciP_Gi²

. 3.12

Computing directly, we have

FPG≡min

ζ∈U

λ ζ_T

P_G−^N^G

i1

a_i b_iP_Gi c_iP_Gi²

λ_T

P_G−^N^G

i1

minζ∈U ζ^TP_G.

3.13

We will use the duality theorem of linear programming to analyze the term min_ζ∈Uζ^TPG. Define the corresponding Lagrangian function as

L

ζ, z, δ, γ

ζ^TP_G−ze^Tζ δ^T ζ−ζ

γ^T ζ−ζ

. 3.14

It holds that

∇ζL

ζ, z, δ, γ

PG−ze−δ γ0, 3.15

which follows

z,δ,γmaxmin

ζ L

ζ, z, δ, γ max

z,δ,γ

δ^Tζ−γ^Tζ:PG−ze−δ γ0, δ≥0, γ ≥0

. 3.16

From the duality theorem of linear programming, we have minζ∈U ζ^TPG max

z,δ,γmin

ζ L

ζ, z, δ, γ

. 3.17

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Then3.13can be reformulated as FPG

max

z,δ,γ

λT

PG−^N^G

i1

ai biPGi ciP_Gi²

δ^Tζ−γ^Tζ

:PG−ze−δ γ0, δ≥0, γ ≥0

. 3.18 Finally, problem3.10with a box uncertainty can be reformulated as

maxPG

ζ∈UinffPG, λ max

PG,z,δ,γ

λ_T

PG−^N^G

i1

ai biPGi ciP_Gi²

δ^Tζ−γ^Tζ

s.t. P_G∈Π,

PG−ze−δ γ0, δ≥0, γ ≥0.

3.19

Remark 3.2. The reformulation 3.19is an ordinary quadratic nonlinear programming ifΠ is linear with respect toP_G, which can be solved easily by many solution methods.

3.4. Counterpart of RO-Based Self-Scheduling with Ellipsoidal Uncertainty

Similarly to the box uncertainty, we define a function as

FPG≡min

ζ∈U

λ Aζ_T

PG−^N^G

i1

ai biPGi ciP_Gi²

λ_T

PG−^N^G

i1

ai biPGi ciP_Gi²

minζ∈U ζ^TA^TPG.

3.20

Consider the term

minζ∈U ζ^TA^TPG 3.21

withU{ζ|e^TAζ0, ζ ≤1}. The correspondent Lagrangian function of3.21is L

ζ, μ, ν

ζ^TA^TP_G−μ e^TAζ

ν

ξ^Tξ−1

withν≥0. 3.22

Note that we use a relation ofζ ≤1⇔ζ^Tζ≤1. From the optimal condition, we have

∇ζL ζ, μ, ν

A^TPG−μA^Te νζ0, e^TAζ0, ζ ≤1, ν≥0. 3.23

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This indicates that at the optimal pointζ, μ, ν, it holds that

minζ∈U L ζ, μ, ν

A^TP_G−μA^Te νζT

ζ−ν−ν. 3.24

Define an auxiliary variableσand letσ≡νζ; it holds that

∇ζL ζ, μ, ν

A^TP_G−μA^Te σ0. 3.25

On the other hand, fromζ ≤1 we have

ν²ζ²σ² ≤ν²⇒ σ ≤ν. 3.26

Note that above relationship involvesν≥0.

Combining with3.23–3.26and the duality theorem of convex programming, we obtain

minζ∈U ζ^TP_Gmax

μ,νmin

ζ∈U L ζ, μ, ν

max

μ,ν,σ

−ν:A^TP_G−μA^Te σ0, σ ≤ν

. 3.27

Therefore, the RO-based self-scheduling with ellipsoidal uncertainty is reformulated as

maxPG inf

ζ∈UfP_G, λ max

PG,μ,ν,σ

λT

P_G−^N^G

i1

−ν

s.t. PG∈Π,

A^TP_G−μA^Te σ0, σ ≤ν.

3.28

Remark 3.3. The reformulation3.28is a quadratic cone programming if the feasible regionΠis linear with respect to variablePG.

Two reformulations3.19and3.28are typical determinate optimization problems, which can be solved by many eﬀective algorithms see 18. Furthermore, except the uncertainty with respect to priceλ, other data can be chosen as uncertain variable, such asi the cost coeﬃcients of generatorsai, b_i, c_i,iithe forecasted power demand at busesP_Di, iiithe bound of variables in constraints.

4. Numerical Examples for Self-Scheduling

In order to validate the RO-based self-scheduling approach, this section provides numerical examples. Some comparing approach with paper3are also done, and the sensitivity with respect to the uncertainty set is analyzed.

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Table 1: Given data and estimate LMPs.

Bus P_G^min P_G^max a b c λⁿ

no. MW MW $/h $/MWh $/MW²h $/MWh

1 50 200 0 2 0.00375 3

2 20 80 0 1.75 0.00175 3

5 15 50 0 1 0.0625 3

8 10 35 0 3.25 0.00834 5

11 10 40 0 3 0.025 5

13 12 40 0 3 0.025 5

G G

1 3

2

4 6

5 7

28 8

13

12 16

17

14 18

15 19

20

23 24

22 21

25 26

29 27 10

9 11

30

G G

G

Figure 1: The IEEE 30-bus system.

4.1. Tested System

We choose the same example in3as the tested system, that is, IEEE-30 system with six generator buses, which are bus-1,2,5,8,11,13, respectively, seeFigure 1.

Consider the mathematical model 3.1–3.6. Here we omit the security3.7. The network, load, and generator data for this system are given in17. The coeﬃcients in the cost function and the bound of generation outputs are specified inTable 1, together with the values of forecastnominalLMPs.

The given constantsP_Di andT_ij^max in the DC network model constraints3.5and in the intact network line flow constraints3.6are reported in Tables2and3, respectively.

4.2. Uncertain Set and Algorithm

In the numerical test, we consider the case of uncertain box set, that is,

U

ζ|e^Tζ0, ζ≤ζ≤ζ

. 4.1

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Table 2: DC network model bounds.

Bus no. 1 2 3 4 5 6 7 8 9 10

PDi 0 21.7 2.4 7.6 94.2 0 22.8 30 0 5.8

Bus-no 11 12 13 14 15 16 17 18 19 20

PDi 0 11.2 0 6.2 8.2 3.5 9 3.2 19.5 2.2

Bus-no 21 22 23 24 25 26 27 28 29 30

PDi 17.5 0 3.2 8.7 0 3.5 0 0 2.4 10.6

Table 3: Intact network line flow.

fbus 1 1 2 3 2 2 4 5 6 6 6

tbus 2 3 4 4 5 6 6 7 7 8 9

T_ij^max 130 130 65 130 130 65 90 70 130 32 65

fbus 6 9 9 4 12 12 12 12 14 16 15

tbus 10 11 10 12 13 14 15 16 15 17 18

T_ij^max 32 65 65 65 65 32 32 32 16 16 16

fbus 18 19 10 10 10 10 21 15 22 23 24

tbus 19 20 20 17 21 22 22 23 24 24 25

T_ij^max 16 32 32 32 32 32 32 16 16 16 16

fbus 25 25 28 27 27 29 8 6

tbus 26 27 27 29 30 30 28 28

T_ij^max 16 16 65 16 16 16 32 32

The fluctuation bound of price is set as a ratio of the estimated priceλ, that is, let

ζn%λ, ζ−ζ. 4.2

Herenis the ratio constant. For example,n5 means that the fluctuation bound of price is 5% ofλ. In the following tests, we will calculate the RO-based self-schedular by diﬀerentn values.

The RO-based self-scheduling with uncertain price is the model 3.19, which is a typical quadratic programming. Then we test the system by Quadprog file in MATLAB toolbox.

4.3. Computational Results

(1) Optimal profit and output of generations

For casen5, we solve the counterpart of RO and obtain the output of generations and the optimal profit as follows:

P_G1128.09MW, P_G235.00MW, P_G518.09MW, P_G835.00MW, P_G11 30.00MW, P_G13 37.22MW,

fmaxPG 241.4$/h.

4.3

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Table 4: Computing results of two methods.

CRP RO

β fmax n fmax

0.9 216.34 5 214.4

0.95 210.40 6 210.71

0.99 207.43 7 207.05

100 120 140 160 180 200 220 240

Profit of IEEE-30 withn

Profit($/h)

0 5 10 15 20 25 30 35 40

Percentn

Figure 2: Relation of optimal profit andn.

With the diﬀerentnvalues, we obtain the diﬀerent profit. The results are reported inFigure 2.

The curve indicates that the good estimate value of pricei.e., the small fluctuationnwill result in high profit. For the big n, the obtained profit is conservative.We also make some comparison with paper3 and find that whenn is chosen the value between 5.0–7.0, the computing values, are closed for two methods, seeTable 4. Here CRP represents the results in3, and RO indicates the results in this paper.

(2) Optimal output of generators with diﬀerentn

For the diﬀerent choice n, we obtain the diﬀerent output of generators. The optimal self- scheduling of three cases is reported inTable 5. Comparing with the computing results in3 see Table 3 in3, we find that the value of RO method is conservative. This is identical to the theory analysis since the RO-based approach is set up in the worst-case.

(3) Sensitivity analysis of optimal output of generators

In order to test the eﬀect of turbulence valuen, we repeat to solve the RO model by using the diﬀerentn. The computing results for six generators are shown in Figures3,4,5,6,7, and8.

From the results, we can see that for each generator, they have diﬀerent sensitivity.

iThe output PG1 of the slack bus-1 is decreasing when the value n is increasing.

Especially, it has a big decrease forn≥10.

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Table 5: Optimal output of generators.

PGat bus-barMW n5 n10 n15

1 128.09 127.77 125.66

2 35.00 35.00 35.00

5 18.09 20.47 22.74

8 35.00 35.00 35.00

11 30.00 30.00 30.00

13 37.22 35.17 35.00

Output of bus-1 withn

Output(MW/h)

114 116 118 120 122 124 126 128 130

0 5 10 15 20 25 30 35 40

Percentn

Figure 3: Optimal output of bus-1 with diﬀerentn.

Output(MW/h)

35 35 35 35 35 35 35 35 35

350 5 10 15 20 25 30 35 40

Percentn

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Output(MW/h)

16 18 20 22 24 26 28 30 32 34 36

0 5 10 15 20 25 30 35 40

Percentn

Output(MW/h)

35 35 35 35 35 35 35 35 35

350 5 10 15 20 25 30 35 40

Percentn

Figure 6: Optimal output of bus-8 with diﬀerentn.

iiThe outputsP_G2, P_G8, P_G11at bus-2, bus-8, and bus-11 are not sensitive with respect to the variety ofn. This is favorable for the persistent output of the generator, and reduces the times of on-oﬀgenerators.

iiiThe outputPG5 at bus-5 is increasing in a linear version with respect ton, which means that the bus-5 is sensitive.

ivThe outputPG13at bus-13 is decreasing sharply whennfrom 1 to 10. It almost takes on a stability state whenn≥10.

The above analysis can provide some message to generation company for the self- scheduling and then guides the bidding action of generation company.

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Output(MW/h)

0 5 10 15 20 25 30 35 40

Percentn 30

30 30

30 30 30 30 30 30 30

Output(MW/h)

0 5 10 15 20 25 30 35 40

Percentn 34.5

35 35.5 36 36.5 37 37.5 38 38.5 39

Figure 8: Optimal output value of bus-13 with diﬀerentn.

5. Conclusion

This paper presents a new methodology to study the generation of self-scheduling in power market. Based on robust optimizationRO, a new self-scheduling model is established under uncertain price. The counterpart of the model is a quadratic-type programming, which can be solved by many optimization algorithms. IEEE-30 system is chosen as a tested system.

The computing results show that the new method is promising. Comparing our method with other stochastic methodse.g., CVaR approach, the computing result is conservative. From viewpoint of practical applications, the new approach is very suitable for case where the prediction of random variables is diﬃcult. On the other hand, the robust consideration can ensure the security requirements of the systems.

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We just consider the uncertain price with respect to the price-taker in this study. In fact, the proposed method can be extended to other uncertain cases for self-scheduling, such as the price-maker schedular problem and the uncertainty for cost parameters or load demand.

Moreover, other related optimization problems in electricity market can be adopted RO-based approach. For example, the bidding analysis and the optimal power flowOPFwith new energy source. These are worthy problems of our further research.

Acknowledgments

This work is supported by Natural Science Foundation of China10871031, 10926189and the Natural Science united Foundation of Hunan-Hengyang10JJ8008and the Grant of Hunan Education Department10A015.

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